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PREPRINT 1 Nonlinear Model Predictive Control for Constrained Output Path Following Timm Faulwasser Laboratoire d’Automatique, Ecole Polytechnique F´ ed´ erale de Lausanne, EPFL STI IGM LA Station 9, CH-1015 Lausanne, Switzerland. Fon: +41 21 69 37341 Fax: +41 21 69 32574. E-mail: timm.faulwasser@epfl.ch (correspondence address) Rolf Findeisen Institute for Automation Engineering, Otto von Guericke University Magdeburg, Universit¨ atsplatz 2, 39106 Magdeburg, Germany. Fon: +49 391 67 18577 Fax: +49 391 67 11191. E-mail: rolf.fi[email protected]. Abstract—We consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre- specified timing requirements. Such problems are commonly referred to as constrained output path-following problems. Specifically, we propose a predictive control approach to constrained path-following problems with and without velocity assignments and provide sufficient convergence conditions based on terminal regions and end penalties. Furthermore, we analyze the geometric nature of constrained output path-following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions. We draw upon an example from robotics to illustrate our findings. Index Terms—path following, nonlinear model predictive control, stability, constraints, transverse normal forms I. I NTRODUCTION The prototypical problem in control is the stabilization of a set-point. Besides stabilization, the design of controllers for the tracking of time-varying references is also well-understood. Yet not all problems encountered in applications are set-point stabilization or trajectory-tracking problems. One example is the precise steering of a robotic tool along a geometric curve in the robot workspace. Typically, the highest priority is given to the minimization of the deviation between the geometric reference path and the robot tool. The velocity to move along the reference is of secondary interest and might be adjusted in order to achieve better accuracy. Thus neither the stabilization of a set-point nor the tracking of a pre-defined time-varying reference is at the core of this problem. Such control problems—that require to steer a system along a geometric reference curve, whereby the speed along this reference is a degree of freedom in the controller design—are termed path-following problems [1]–[3]. Besides its relevance for applications recent interest in path following is motivated by the fact that in contrast to trajectory tracking, path-following tasks of non-minimum-phase systems are not necessarily subject to fundamental limits of performance [1], [4]. Two approaches to path following have been dominantly dis- cussed in the literature: geometric control design methods and Lyapunov/backstepping techniques, see [5]–[7], respectively, [2], [3], [8]–[13]. The direct consideration of constraints on inputs and/or states, however, is difficult for either approaches. To overcome this limitation nonlinear model predictive control (NMPC) schemes tailored to path-following problems have been proposed. The early works [14], [15] as well as the results presented in [16] are restricted to reference paths in the state space. This limits the applicability, since many realistic path-following problems—e.g. movement tasks for robots, autonomous vehicles, ship, and unmanned Main parts of this research have been conducted while TF was with the Institute for Automation Engineering, Otto-von-Guericke-University Magde- burg, Germany. aerial vehicles—are defined in an output space rather than in the state space. Path following in output spaces is also termed output path following. Successful implementations of predictive output path following to real systems have been reported in [17]–[19]. Predictive output path following for underwater vehicles and non-holonomic systems is discussed in [20] and [21]. 1 Besides practical considerations, the question of stability/convergence is challenging in output path following. First steps in this direction are presented in [20], [25], [26]. While in [26] recursive feasibility is lost due to contraction constraints, the preliminary results in [20], [25] draw upon terminal regions and end penalties to guarantee path convergence. However, a common drawback of these works is that no insight into the structure of constrained output path-following problems is provided. In the present contribution, we extend and generalize previous results on predictive control for output path-following problems. In contrast to [14]–[16], [25], [26], we investigate two different kinds of path-following problems, i.e., with and without velocity assignments for the reference evolution. Similar to [20], [25], we present a continuous-time sampled-data NMPC framework applicable to the design of controllers for output path-following problems under direct consideration of constraints on states and inputs. Sufficient conditions based on terminal regions and end penalties guaranteeing the conver- gence to an output path as well as recursive feasibility of the arising optimization problems are provided. We extend our previous results [25] by investigating the geometric nature of path-following problems for nonlinear systems via the analysis of transverse normal forms and their use for the computation of stabilizing terminal regions and end penalties. This way, we provide a general framework for the design of continuous-time predictive control scheme for constrained output path-following problems. The remainder of the paper is structured as follows. In Section II we outline the considered output path-following problems. Section III contains the main contributions, i.e., a predictive control framework to path-following problems including sufficient convergence conditions. The design of suitable stabilizing terminal regions and end penalties is discussed in Section IV. To support our results we draw upon an example from robotics in Section V. 1 Besides predictive (feedback) control approaches to path-following prob- lems, optimization-based feedforward path following has been in discussed in the literature [?], [22]–[24]. These methods assume a special system structure—usually, it is required that the path is defined in a flat output space of a differentially flat system—and they are restricted to the computation of feedforward or open-loop controls. arXiv:1502.02468v1 [cs.SY] 9 Feb 2015

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PREPRINT 1

Nonlinear Model Predictive Control forConstrained Output Path Following

Timm FaulwasserLaboratoire d’Automatique, Ecole Polytechnique Federale de Lausanne,

EPFL STI IGM LA Station 9, CH-1015 Lausanne, Switzerland.Fon: +41 21 69 37341 Fax: +41 21 69 32574. E-mail: [email protected] (correspondence address)

Rolf FindeisenInstitute for Automation Engineering, Otto von Guericke University Magdeburg,

Universitatsplatz 2, 39106 Magdeburg, Germany.Fon: +49 391 67 18577 Fax: +49 391 67 11191. E-mail: [email protected].

Abstract—We consider the tracking of geometric paths in output spacesof nonlinear systems subject to input and state constraints without pre-specified timing requirements. Such problems are commonly referred toas constrained output path-following problems. Specifically, we proposea predictive control approach to constrained path-following problemswith and without velocity assignments and provide sufficient convergenceconditions based on terminal regions and end penalties. Furthermore,we analyze the geometric nature of constrained output path-followingproblems and thereby provide insight into the computation of suitableterminal control laws and terminal regions. We draw upon an examplefrom robotics to illustrate our findings.

Index Terms—path following, nonlinear model predictive control,stability, constraints, transverse normal forms

I. INTRODUCTION

The prototypical problem in control is the stabilization of aset-point. Besides stabilization, the design of controllers for thetracking of time-varying references is also well-understood. Yet notall problems encountered in applications are set-point stabilizationor trajectory-tracking problems. One example is the precise steeringof a robotic tool along a geometric curve in the robot workspace.Typically, the highest priority is given to the minimization of thedeviation between the geometric reference path and the robot tool.The velocity to move along the reference is of secondary interestand might be adjusted in order to achieve better accuracy. Thusneither the stabilization of a set-point nor the tracking of a pre-definedtime-varying reference is at the core of this problem. Such controlproblems—that require to steer a system along a geometric referencecurve, whereby the speed along this reference is a degree of freedomin the controller design—are termed path-following problems [1]–[3].

Besides its relevance for applications recent interest in pathfollowing is motivated by the fact that in contrast to trajectorytracking, path-following tasks of non-minimum-phase systems arenot necessarily subject to fundamental limits of performance [1],[4]. Two approaches to path following have been dominantly dis-cussed in the literature: geometric control design methods andLyapunov/backstepping techniques, see [5]–[7], respectively, [2], [3],[8]–[13]. The direct consideration of constraints on inputs and/orstates, however, is difficult for either approaches.

To overcome this limitation nonlinear model predictive control(NMPC) schemes tailored to path-following problems have beenproposed. The early works [14], [15] as well as the results presentedin [16] are restricted to reference paths in the state space. This limitsthe applicability, since many realistic path-following problems—e.g.movement tasks for robots, autonomous vehicles, ship, and unmanned

Main parts of this research have been conducted while TF was with theInstitute for Automation Engineering, Otto-von-Guericke-University Magde-burg, Germany.

aerial vehicles—are defined in an output space rather than in thestate space. Path following in output spaces is also termed outputpath following. Successful implementations of predictive output pathfollowing to real systems have been reported in [17]–[19]. Predictiveoutput path following for underwater vehicles and non-holonomicsystems is discussed in [20] and [21].1

Besides practical considerations, the question ofstability/convergence is challenging in output path following.First steps in this direction are presented in [20], [25], [26]. Whilein [26] recursive feasibility is lost due to contraction constraints,the preliminary results in [20], [25] draw upon terminal regions andend penalties to guarantee path convergence. However, a commondrawback of these works is that no insight into the structure ofconstrained output path-following problems is provided.

In the present contribution, we extend and generalize previousresults on predictive control for output path-following problems. Incontrast to [14]–[16], [25], [26], we investigate two different kinds ofpath-following problems, i.e., with and without velocity assignmentsfor the reference evolution. Similar to [20], [25], we present acontinuous-time sampled-data NMPC framework applicable to thedesign of controllers for output path-following problems under directconsideration of constraints on states and inputs. Sufficient conditionsbased on terminal regions and end penalties guaranteeing the conver-gence to an output path as well as recursive feasibility of the arisingoptimization problems are provided. We extend our previous results[25] by investigating the geometric nature of path-following problemsfor nonlinear systems via the analysis of transverse normal forms andtheir use for the computation of stabilizing terminal regions and endpenalties. This way, we provide a general framework for the designof continuous-time predictive control scheme for constrained outputpath-following problems.

The remainder of the paper is structured as follows. In Section IIwe outline the considered output path-following problems. Section IIIcontains the main contributions, i.e., a predictive control framework topath-following problems including sufficient convergence conditions.The design of suitable stabilizing terminal regions and end penaltiesis discussed in Section IV. To support our results we draw upon anexample from robotics in Section V.

1 Besides predictive (feedback) control approaches to path-following prob-lems, optimization-based feedforward path following has been in discussedin the literature [?], [22]–[24]. These methods assume a special systemstructure—usually, it is required that the path is defined in a flat output spaceof a differentially flat system—and they are restricted to the computation offeedforward or open-loop controls.

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Notation

The point-wise image of a set X ⊂ Rnx under a map h : Rnx →Rny is denoted as h(X ) := {y ∈ Rny |x ∈ X 7→ y = h(x)}.The interior and the boundary of a set X are denoted as int(X ),respectively, ∂X . An open neighborhood of a point x ∈ Rnx isdenoted asNx. The kth time derivative of a function r : [t0,∞)→ Ris written as dkr(t)

dtkor more conveniently r(k). Ck denotes the set

of k-times continuously differentiable functions. The set of piece-wise continuous and right continuous functions on R that take valuesin V ⊂ Rm is shortly denoted as PC(V). The norm ‖x‖ ofx ∈ Rnx denotes the 2-norm. For Q ∈ Rn×n ‖x‖2Q = xTQx,while ‖Q‖ denotes the induced 2-norm and ‖x‖∞ denotes theinfinity norm. The identity matrix of Rnx is written as Inx . We

use Inx :=

(0nx−1,1 Inx−1

0 01,nx−1

)and Enx := (0, . . . , 0, 1)T ∈

Rnx . A = diag(a1, a2, . . . , anx) denotes a diagonal matrix withentries a1, . . . , anx .

The solution of an ordinary differential equation x = f(t, x, u),starting at time t0 at x(t0) = x0 driven by an input u : [t0,∞) →Rnu , is written as x(·, t0, x0|u(·)). The value of this solution attime t1 ≥ t0 is denoted as x(t1, t0, x0|u(·)). The total derivativeof a function E(t, x(t)) ∈ C1 with respect to t is written asd

dt

[E(t, x(t))

]:=

∂E

∂t+∂E

∂xx(t). The evaluation of E (t, x(t))

at t = t1 + T is written as E (t, x(t))|t=t1+T .

II. PATH-FOLLOWING PROBLEMS

We consider nonlinear systems of the form

x = f(x) +

nu∑j=1

gj(x)uj , x(t0) = x0 (1a)

y = h(x). (1b)

The map h : Rnx → Rny (1b) defines the output y ∈ Rnyor the variables of specific interest.. We assume that the mapsf : Rnx → Rnx , gj : Rnx → Rnx , h : Rnx → Rny aresufficiently often continuously differentiable. Here x ∈ X ⊆ Rnxand u ∈ U ⊂ Rnu denote the closed set of state constraints and thecompact set of input constraints.

Set-point stabilization usually refers to the task of stabilizing afixed point in the state space. Trajectory tracking requires conver-gence of the states or the outputs of a system to a time-dependentreference that implies an explicit requirement when to be where onthe reference.2 In contrast to trajectory-tracking problems we aimat driving the system along a geometric reference without any pre-specified timing information. This geometric reference is denoted aspath P . We assume it is given as a parametrized regular curve in theoutput space (1b)

P = {y ∈ Rny | θ ∈ [θ0, θ1] 7→ y = p(θ)} . (2)

Here the scalar variable θ is called the path parameter and p : R→Rny is called a parametrization of P . Note that the regularity of ageometric curve implies the local bijectivity of the parametrizationp(θ), cf. [30]. The map p : R → Rny is assumed to be sufficientlyoften continuously differentiable. In general, the path parameter θ istime dependent but its time evolution t 7→ θ(t) is not known a priori.

2Note that in the literature different terminologies are used for trajectory-tracking problems. For instance, if the task is to track a trajectory defined in anoutput space and the reference trajectory is generated by an exogenous system(or exo-system), then one refers to the problem either as model-followingproblem, servo problem or as output regulation problem, cf. [27], [28]. Wefollow along the classic lines of [29] and deliberately denote all these casesas trajectory-tracking problems.

Subsequently, path following refers to the problem of steering theoutput (1b) to the path P and to follow it along in direction ofincreasing values of θ. Obviously, one could solve this problem bychoosing a fixed timing θ(t) and designing a trajectory-tracking con-troller for p(θ(t)). This way, path following would be reformulatedas a trajectory-tracking problem. However, the degree of freedom ofadjusting θ(t) is lost. Here, we tackle the problem differently. Theconceptual idea is to obtain the system input u : [t0,∞) → U andthe reference timing t 7→ θ(t) in the controller, i.e., the controllerdetermines the input u(t) to converge to reference path as well asthe time evolution θ(t) of the reference. In other words, we considerthe following problem:

Problem 1 (Constrained output path following): Given the system(1) and the reference path P (2), design a controller that computesu(t) and θ(t) and achieves:

i) Path convergence: The system output y = h(x) converges to theset P in the sense that

limt→∞‖h(x(t))− p(θ(t))‖ = 0.

ii) Convergence on path: The system moves along P in forwarddirection, i.e.

θ(t) ≥ 0 and limt→∞‖θ(t)− θ1‖ = 0.

iii) Constraint satisfaction: The constraints on the states x(t) ∈ Xand the inputs u(t) ∈ U are satisfied for all times.

Sometimes it might be desired to track a speed profile along thepath. Following along the lines of [1], [4], [12] such a problem is de-noted as constrained output path following with velocity assignment.It differs from Problem 1 in part ii):

Problem 2 (Output path following with velocity assignment):Given the system (1) and the reference path P (2), design acontroller that computes u(t) and θ(t), achieves part i) & iii) ofProblem 1 and guarantees:

ii) Velocity convergence: The path velocity θ(t) converges to apredefined profile such that

limt→∞‖θ(t)− θref (t)‖ = 0.

Note that path following with velocity assignment is not equivalentto trajectory tracking, since path following with speed assignmentdoes in general not specify a unique output reference p(θ(t)). Ratherit admits several reference trajectories p(θi(t)), i ∈ {1, 2, . . . }, withθi(t) = θref (t), which may differ with respect to θ, i.e., θi(t) 6=θj(t), i 6= j.

A classical design of path-following controllers regards the pathparameter as a virtual state, whose evolution is determined throughan additional ordinary differential equation (ODE) denoted as timinglaw. In essence, the timing law is an additional degree of freedom inthe controller design. In backstepping approaches to path following,for instance, this timing law is constructed such that path convergenceis enforced [3], [12]. For sake of simplicity we use a simple integratorchain as timing law, i.e., the timing of the path parameter θ isspecified via the ODE

θ(r) = v, θ(i)(t0) = θ(i)0 , i = 0, . . . , r − 1, (3)

where, depending on the value of r, the variable v can be regarded asthe speed, acceleration or jerk of the reference. It is crucial to note thatthe time evolution θ(t)—and thus also the evolution of the referencep(θ(t))—can be controlled via the virtual input v : [t0,∞) → V .At this point we do not specify the length r ∈ N of the integratorchain (3), which will depend on the design method and the systemconsidered. We will come back to this issue in Section IV.

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Relying on the timing law (3) we suggest to tackle path-followingproblems via the augmented system description

x = f(x) +

nu∑j=1

gj(x)uj , (4a)

z = I rz + Erv (4b)

e = h(x)− p(z1), (4c)

θ = z1. (4d)

Here, (4a) includes the dynamics of the system to be controlled(1), (4b) is the timing law (3) with z = (θ, θ, . . . , θ(r−1))T . Theerror output (4c) represents the deviation from the path, while (4d)describes the current reference position on the path.

III. MODEL PREDICTIVE PATH-FOLLOWING CONTROL

Subsequently, we propose a predictive path-following controlscheme to tackle path-following problems. We denote this schemeas model predictive path-following control (MPFC). We will firstfocus the investigations on Problem 1, including the presentation ofsufficient convergence conditions. The extension to path followingwith speed assignment (Problem 2) is discussed at the end of thissection.

A. Proposed Predictive Control Scheme

As standard in predictive control the applied input is based onrepeatedly solving an optimal control problem (OCP). That is, ateach sampling instance tk = kδ, k ∈ N0, δ > 0 we solve an OCPthat minimizes the cost functional

J (x(tk), z(tk), uk(·), vk(·))

=

∫ tk+T

tk

F(e(τ), θ(τ), uk(τ), vk(τ)

)dτ

+ E (t, x(t), z(t))|t=tk+T . (5)

As usual in NMPC the function F : Rny × R × V × U → R+0 is

termed cost function, and E : R+0 × Rnx × Rr → R+

0 is denotedas terminal or end penalty; predicted states and inputs are indicatedby the superscript ·. The subscript ·k indicates that an open-loopinput uk(·) is computed at the kth sampling instant tk. The constantT ∈ (δ,∞) is called the prediction horizon. The OCP to be solvedin a receding horizon fashion at the sampling times tk reads:

minimize(uk(·),vk(·))∈PC(U×V)

J (x(tk), z(tk), uk(·), vk(·)) (6a)

subject to ∀τ ∈ [tk, tk + T ] :

˙x(τ) = f(x(τ)) +

nu∑j=1

gj(x(τ))uk,j(τ)), x(tk) = x(tk)

(6b)˙z(τ) = I rz(τ) + Ervk(τ) (6c)

z(tk) = z(tk, tk−1, z(tk−1)|v?k−1(·)) (6d)

e(τ) = h(x(τ))− p(z1(τ)) (6e)

θ(τ) = z1(τ) (6f)

x(τ) ∈ X , uk(τ) ∈ U (6g)

z(τ) ∈ Z, vk(τ) ∈ V (6h)

(x(tk + T ), z(tk + T ))T ∈ E ⊂ X × Z. (6i)

For sake of simplicity we assume that an optimal solution to OCP(6) exists and is attained. The statement of conditions, which ensurethe existence of optimal solutions is beyond the scope of this paper,

Data: x(t0), z(t0), δ, T

Step 0: Initialize k = 0.Step 1: Get state information x(tk).Step 2: Solve OCP (6) with initial condition x(tk), z(tk).Step 3: Apply optimal input

∀t ∈ [tk, tk + δ) : u(t) = u?k(t).

Step 4: Assign z(tk+1) = z(tk+1, tk, z(tk)|v?k(·)).Step 5: k → k + 1 Goto Step 1.

Fig. 1: MPFC scheme based on OCP (6).

instead we refer to [31], [32]. Note that the decision variables ofthe minimization in (6a) are the real system input u(·) ∈ PC(U)as well as the virtual path parameter input v(·) ∈ PC(V). In otherwords, by solving (6) we obtain the system input and the referenceevolution at the same time. State and input constraints of the system tobe controlled are enforced by (6g). Furthermore, the path parameterdynamics (6c) are subject to the state and input constraints (6h),whereby the state constraint Z is defined as

Z := [θ0, θ1]× R+0 × Rr−2 ⊂ Rr. (7)

Essentially, this constraint ensures that θ = z1 ∈ [θ0, θ1], as wellas ˙θ ≥ 0. This way we enforce monotonous forward motion alongthe path. In order to avoid impulsive solutions of the path parameterdynamics (6c) the admissible values of the virtual path parameterinputs v are restricted to a compact set V ⊂ R containing 0 in itsinterior in (6h).

While at each sampling instance the measured state informationx(tk) serves as initial condition for (6b), the initial conditionof the timing law (6c) is based on the last predicted trajectoryz(·, tk−1, z(tk−1)|v?k−1(·)) evaluated at time tk. In cases where noinitial condition for the first sampling instance k = 0 is given, weobtain z(t0) via

z(t0) = (θ(t0), 0, . . . , 0)T (8a)

θ(t0) = argminθ∈[θ0,θ1]

‖h(x0)− p(θ)‖. (8b)

In general, this problem might have multiple optimal solutions, andwe simply choose one of them.

Similar to classical NMPC schemes [33]–[35] the terminal con-straint (6i) enforces that at the end of each optimization the predictedaugmented state (x(tk + T ), z(tk + T ))T lies inside a terminalregion E ⊆ X × Z . Although only outputs and inputs are penalizedin the cost function F in (5), the terminal constraint is stated inthe state space. The reason for this choice is that—under suitableassumptions—output path following can be reformulated as a man-ifold stabilization problem in the state space, cf. [7], [36]. We willinvestigate this issue in detail in Section IV. Also note that theterminal penalty E will be used to obtain an upper bound on thecost associated to solutions originating inside the terminal regionE ⊆ X × Z . Thus E is stated as a function of the augmented state(x, z)T . Additionally, and without loss of generality, we considerexplicit time dependence of E in (5).

The optimal solution of (6) is denoted asJ? (x(tk), z(tk), u?k(·), v?k(·)). It is specified by optimal inputtrajectories u?k : [tk, tk + T ] → U and v?k : [tk, tk + T ] → V . Now,we are ready to summarize the MPFC scheme in Figure 1. As usualin NMPC, in Step 1 we need to obtain (observed or measured) stateinformation. In Step 2 we solve the OCP (6). And in Step 3, thefirst part of the optimal input u?k(·) is applied to the real system (1)

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until the next sampling time. Note that the virtual input v?k(·) andthe path parameter state z(·) are merely internal controller variables,i.e., in Step 4 the next iteration is prepared.

Remark 1 (Dynamic nature of the MPFC scheme): It should benoted that the solution to (6) at time tk depends on the solutionat the previous sampling instant tk−1. The reason is that the initialcondition of z at time tk, k > 0 is based on the last predictedtrajectory z(·, tk−1, z(tk−1) | v?k(·)) evaluated at time tk, cf. (6d)and Step 4 shown in Figure 1. In other words, the path parameterstate z is as an internal state of the MPFC scheme. Thus, in contrast tousual NMPC schemes for set-point stabilization such as [33]–[35],the MPFC scheme as depicted in Figure 1 is a dynamic feedbackstrategy.

Remark 2 (Computational demand): We remark that the presentpaper is focused on the concept of predictive path following andits properties. Thus, the efficient numerical implementation of theproposed scheme is beyond its scope. However, note that (6) is atypical OCP for an NMPC scheme with terminal constraints andterminal penalties. The only difference compared to NMPC for set-point stabilization are the increased state and input dimensions. Thus,to solve (6) one may apply existing numerical tools tailored for real-time feasible NMPC with state and terminal constraints, cf. also thesuccessful implementations of NMPC for path following in [17]–[19].

B. Sufficient Convergence Conditions

As is well known the receding horizon application of optimal open-loop inputs does not necessarily lead to stability nor to convergence ofthe closed-loop output to the path [34], [35]. Thus we are interestedin conditions ensuring that the MPFC scheme (6) solves Problem1. In order to present such conditions we rely on the followingassumptions.

Assumption 1 (System dynamics): The vector fields f : Rnx →Rnx and gj : Rnx → Rnx , j = 1, . . . , nu from (1) are continuousand locally Lipschitz for any pair (x, u)T ∈ X × U .

Assumption 2 (Continuity of system trajectories): For any x0 ∈X and any input function u(·) ∈ PC(U) the system (1) has anabsolutely continuous solution.

Assumption 3 (Consistency of path and state constraints): Thepath P from (2) is contained in the interior of the point-wise imageof the state constraints X under the output map h : Rnx → Rnyfrom (1b), i.e., P ⊂ int(h(X )).

Assumption 4 (Cost function): The cost function F : Rny × R ×V×U → R+

0 is continuous. Furthermore, we assume that F is lowerbounded by a class K function, i.e., ψ(‖e, θ− θ1‖) ≤ F (e, θ, u, v).3

Assumptions 1-2 are very similar to the ones made for NMPCfor set-point stabilization problems, cf. [33], [34]. Basically, theseassumptions are used to guarantee the local existence and uniquenessof solutions of (1). Assumption 2 is made in order to apply Barbalat’sLemma in a crucial step of the proof of Theorem 1. Assumptions 3–4are specific for model predictive path-following control. The formeris necessary to avoid cases for which parts of the path are inconsistentwith the state constraints. The latter assumption requires that the costfunction is lower bounded in terms of the path-following error andthe path parameter. This way, we enforce path convergence as well as

3In essence one could write F more general as a function of x, z, u and v.Here, we focus explicitly on cost functions depending on e and θ to highlightthat we do not consider a set-point stabilization problem but a more general(path-following) problem whereby merely outputs are penalized in the costfunction.

convergence on the path. Under the above assumptions the followingresult is obtained.

Theorem 1 (Convergence of MPFC): Consider Problem 1 andsuppose that Assumptions 1–4 hold. Suppose that a terminal regionE ⊂ X × Z and a terminal penalty E(t, x, z) exist such that thefollowing conditions are satisfied:

i) The set E is compact. E(t, x, z) is C1 and positive semi-definitewith respect to (t, x, u).

ii) For all t ∈ [t0,∞) and all (x, z)T ∈ E there exists a scalarε ≥ δ > 0 and admissible inputs (uE(·), vE(·)) ∈ PC(U × V)such that for all τ ∈ [t, t+ δ]

d

[E(τ, x(τ), z(τ))

]+ F

(e(τ), θ(τ), uE(τ), vE(τ)

)≤ 0,

(9)and the solutions x(τ) = x(τ, t, x|uE(·)) and z(τ) =z(τ, t, z|vE(·)), starting at (x, z)T ∈ E , stay in E for allτ ∈ [t, t+ δ].

iii) The OCP (6) is feasible for t0.

Then the MPFC scheme depicted in Figure 1 solves Problem 1.Proof: In essence the proof of this result can be obtained via a

reformulation of the standard results on convergence of continuoustime NMPC for set-point stabilization, see e.g. [33]–[35]. Thus weprovide only a shortened proof here outlining the main differencesto [34].

Step 1 (Recursive feasibility): In the first step recursive feasibilityis shown via the usual concatenation of optimal inputs (u?k(·), v?k(·))with the terminal controls (uE(·), vE(·)) as in [34]. Since these con-catenated inputs ensure positive invariance of the terminal constraintE it immediately follows that the MPFC scheme based on (6) isrecursively feasible.

Step 2 (Constraint satisfaction and forward motion): In the secondstep we verify that ii)-iii) of Problem 1 are satisfied. Recall thatthe terminal constraint set is contained in the state constraints, i.e.,E ⊂ X × Z . Thus part iii) of Problem 1 is satisfied. Furthermore,for all (x, z)T ∈ E we have z ∈ Z from (7), which implies that alsothe forward motion requirement θ = z2 ≥ 0 holds. Hence part ii) ofProblem 1 is ensured.

Step 3 (Path convergence and convergence on path): It remainsto verify that path convergence (part i) of Problem 1) is guaranteed.This is done in the third step. First, we consider the value functionof OCP (6)

V (tk, x(tk), z(tk)) := J (x(tk), z(tk), u?k(·), v?k(·)) .

Similar to [34, Lemma 5] one uses the invariance condition (9) toshow that for all sampling times δ ∈ (0, ε] we have

V (tk+1, x(tk+1), z(tk+1))− V (tk, x(tk), z(tk))

≤ −∫ tk+1

tk

ψ(‖e(t), θ(t)− θ1‖)dt.

Second, we consider the MPC value function

V δ(t, x(t), z(t)) :=

V (tk, x(tk), z(tk))−∫ t

tk

F (e(τ), θ(τ), u?k(τ), v?k(τ))dτ,

which is the remainder of V (tk, x(tk), z(tk)) for x(t) =x(t, tk, x(tk)|u?k(·)) and z(t) = z(t, tk, z(tk)|v?k(·)). In the def-inition of V δ(t, x(t), z(t)) the time instant is tk = kδ withk = max

k∈N{k | tk ≤ t}, i.e., the closest previous sampling instant.

One can apply the same ratio as in [34, Lemma 6] to show that for

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all t ≥ t0 it holds that

V δ(t, x(t), z(t)) +

∫ t

t0

ψ(‖e(τ), θ(τ)− θ1‖)dτ

≤ V δ(t0, x(t0), z(t0)).

Finally, we use Assumption 2 and apply Barbalat’s Lemma [37,Lemma 4] to establish convergence lim

t→∞‖e(t), θ(t)− θ1‖ = 0. This

finishes the proof.

Note that the proposed control scheme aims on convergence ofthe output y = h(x) to the path and not on Lyapunov-like statestability.4 In other words, Theorem 1 allows cases where the outputconverges to the path while the states might move through X × Z .This means that general cases, in which the internal dynamics of(4) with respect to the output (e, θ)T are merely bounded in X ×Z but not asymptotically convergent, are possible. At the end ofeach finite prediction horizon, however, the predicted states have toreach the terminal constraint E ⊂ X × Z . This implies that in thenominal case without plant-model mismatch all states of (4)—whichincludes the states of the zero dynamics of (4) with respect to theoutput (e, θ)T—are bounded. Thus the fact that we merely penalizeoutputs in the cost function F does not lead to further difficulties. Itis also straightforward to see that Theorem 1 holds for the special,and usually hardly application relevant, case of invertible output mapsh : Rnx → Rnx and for paths directly defined in the state space.

Remark 3 (Non-input-affine systems): We point out that the proofof Theorem 1 does not rely on the specific input-affine structure of thesystem (1). Indeed, the conditions of the theorem hold even for casesof non-input-affine systems, i.e., general systems of the form x =f(x, u), y = h(x). The main reason to consider input-affine systemsis that this choice allows further insight into the geometric nature ofpath-following problems. And, as we will show subsequently, thisrestriction of the considered system class simplifies the computationof end penalties and terminal regions satisfying the conditions ofTheorem 1.

C. Extension to Predictive Path Following with Velocity Assignment

At this point it is fair to ask how the result of Theorem 1 can beextended to velocity-assigned path following as described in Problem2. To this end we modify Assumption 4 as follows:

Assumption 5 (Cost function): The cost function F : Rny × R ×V×U → R+

0 is continuous. Furthermore, we assume that F is lowerbounded by a class K function, i.e., ψ(‖e, θ− θref‖) ≤ F (e, θ, u, v).

For velocity-assigned path-following problems the path parameterθ = z1 might grow unbounded. Thus the constraint z ∈ Z , cf. (6h),on the path parameter states should be dropped. The next result statesthat a modified MPFC scheme, in which a cost function accordingto Assumption 5 and no path parameter state constraint (Z = Rr)are considered, solves Problem 2.

Theorem 2 (Convergence of MPFC with velocity assignment):Consider Problem 2 and suppose that Assumptions 1–3 and 5 hold.Suppose that Z = Rr , a terminal region E ⊂ X × Rr and aterminal penalty E(t, x, z) exist such that the following conditionsare satisfied:

i) The set E is compact with respect to x and closed with respectto z. E(t, x, z) is C1 with respect to (t, x, u) and positive semi-definite.

4Even for sampled-data continuous-time NMPC tailored to set-point stabi-lization it is in general difficult to prove Lyapunov stability. Usually, merelyasymptotic convergence is established [34]. This is due to the fact that betweentwo sampling instances tk and tk+1 the controller applies open-loop inputsto the system.

ii) For all t ∈ [t0,∞) and all (x, z)T ∈ E there exist a scalarε ≥ δ > 0 and admissible inputs (uE(·), vE(·)) ∈ PC(U × V)such that for all τ ∈ [t, t+ δ]

d

[E(τ, x(τ), z(τ))

]+ F

(e(τ), θ(τ), uE(τ), vE(τ)

)≤ 0,

(10)and the solutions x(τ) = x(τ, t, x|uE(·)) and z(τ) =z(τ, t, z|vE(·)), starting at (x, z)T ∈ E , stay in E for allτ ∈ [t, t+ δ].

iii) The OCP (6) is feasible for t0.Then the MPFC scheme depicted in Figure 1 solves Problem 2.

The proof of this result is similar to the proof of Theorem1. Recursive feasibility and constraint satisfaction can be shownalong the same lines. The only difference is that in the last stepthe application of Barbalat’s Lemma leads to the conclusion thatlimt→∞

‖e(t), θ(t)− θref (t)‖ = 0.

IV. DESIGN OF SUITABLE TERMINAL REGIONS

AND END PENALTIES

So far we have shown that a suitable combination of a terminalregion and an end penalty can be used to guarantee convergenceof predictive path following. Similar to the case of NMPC forstabilization and tracking problems, [33], [38], [39], the designof terminal regions and corresponding end penalties is challengingfor the proposed MPFC scheme. In general, the computation ofterminal regions involves the design of a locally admissible controller.Subsequently, we present two technical results that allow using trivialend penalties E(t, x(t), z(t)) = 0. And later we discuss the inherentgeometric properties of path-following problems.

A. Trivial End Penalties

As a preparation step we introduce the notation

ϕE(t) := F (e(t), θ(t), uE(t), vE(t))

describing the evolution of the cost function for given inputs(uE(·), vE(·)) ∈ PC(U × V).

Lemma 1 (Existence of a time-dependent terminal penalty): As-sume that there exist terminal controls (uE(·), vE(·)) ∈ PC(U × V)defined for all t ∈ [t0,∞) and a compact terminal region E ⊂ X×Zsuch that the following conditions hold:

i) The set E ⊂ X×Z is rendered controlled positively invariant by(uE(·), vE(·)), i.e., any solution x(t) = x(t, t0, x |uE(·)) andz(t) = z(t, t0, z | vE(·)), starting at time t0 at any (x, z)T ∈ E ,stays in E for all t ∈ [t0,∞).

ii) For all (x, z)T ∈ E and all t ≥ t0 it holds that

ϕE(t) ≤ c(x, z)e−α(x,z)(t−t0) (11)

and for all (x, z)T ∈ E :

0 < α ≤ α(x, z) and 0 ≤ c(x, z) ≤ c <∞.

Then, there exists an end penalty E : [t0, ∞) → R+\∞, such thatE and E satisfy the conditions of Theorem 1.

Proof: Without difficulties if follows from part ii) of the lemmathat for all (x, z)T ∈ E and all t ∈ [t0,∞)

ϕE(t) ≤ c(x, z)e−α(x,z)(t−t0) ≤ c(x, z)e−α(t−t0) ≤ ce−α(t−t0).

It is easy to verify that for all (x, z)T ∈ E

E(t) = cα−1e−α(t−t0) (12)

satisfies the cost decrease condition (9).

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Now, consider two variants of the objective functional (5) differingonly by the end penalty

J1 (uk(·), vk(·)) =

∫ tk+T

tk

F(e(τ), θ(τ), uk(τ), vk(τ)

)dτ,

J2 (uk(·), vk(·)) =

∫ tk+T

tk

F(e(τ), θ(τ), uk(τ), vk(τ)

)dτ

+ E(tk + T ).

Note that for sake of simplified notation we neglect the dependenceof Ji, i ∈ {1, 2} on x(tk), z(tk). We denote two variants of OCP(6) as follows: (6) with J1 is denoted as OCP1 and (6) with J2 isdenoted as OCP2.

Lemma 2 (Equivalence of optimal solutions): Consider OCPi,i ∈ {1, 2} subject to the same initial condition x(tk), z(tk). Thefollowing two statements hold:

i) Suppose that OCP1 has an optimal solution u?k(·), v?k(·), thenu?k(·), v?k(·) is also an optimal solution to OCP2.

ii) Suppose that OCP2 has an optimal solution u?k(·), v?k(·), thenu?k(·), v?k(·) is also an optimal solution to OCP1.Proof: We first consider statement i). For any admissible choice

of uk(·), vk(·) it holds that

J2 (uk(·), vk(·))− J1 (uk(·), vk(·)) = E(tk + T ).

This means that, for any sampling instant tk and any initial conditionx(tk), z(tk), the two objective functionals Ji, i ∈ {1, 2} only differby a constant. And since E(t) from (12) depends only on t andnot on x or z, the value of E(tk + T ) is not influenced by thechoice of uk(·), vk(·). Hence, any input u?k(·), v?k(·), which is anoptimal solution to OCP1, is also an optimal solution to OCP2. Theproof of statement ii) is obtained without difficulties based on similararguments.The next result shows how the last two lemmas can be combined.

Proposition 1 (Trivial end penalty E(t, x(t), z(t)) = 0):Suppose that there exist terminal controls (uE(·), vE(·)) ∈ PC(U ×V) defined for all t ∈ [t0,∞) and a compact terminal region E ⊂X ×Z such that conditions i)–ii) of Lemma 1 hold. Then the MPFCscheme using the end penalty E(t, x(t), z(t)) = 0 and the terminalregion E solves Problem 1.

Proof: From Lemma 1 we know that E and E satisfy theconditions of Theorem 1, i.e., they enforce path convergence andconvergence on the path. From Lemma 2 we know that usingE(t) = 0 instead of E(t) in OCP (6) we obtain inputs that areoptimal for OCP (6) with E(t). Thus, applying the end penaltyE(t, x(t), z(t)) = 0 combined with the terminal region E , we obtainthe conclusions of Theorem 1.

Remark 4 (Exponentially stabilizing terminal controls laws):The main insight obtained by the last proposition can be summarizedas follows: from the stability point of view, terminal controls, whichensure exponential cost decrease in the sense of Lemma 1, renderit unnecessary to determine terminal penalties. However, suitablychosen terminal penalties, can improve closed-loop performance.Thus, it is not surprising that the last proposition can be adjusted toother NMPC schemes designed for stabilization or trajectory tracking,cf. [39].

B. Geometric Structure of Path-following Problems

Next, we show that path-following problems are equivalent to theproblem of stabilizing a certain manifold in the state space, see also[7], [12], [36]. For sake of simplicity the considerations focus onconstrained path following without velocity assignment (Problem 1).Corresponding results can be also established for path following with

velocity assignment (Problem 2). To this end we restrict the class ofconsidered systems (1).

Assumption 6 (Vector relative degree): System (1) has a squareinput-output structure, i.e., dimu = nu = ny = dim y holds, and ithas a well-defined vector relative degree

r = (r1, . . . , rny )T , r = max{r1, . . . , rny}, ρ =

ny∑i

ri (13)

on a sufficiently large set X ⊆ X ⊆ Rnx .Readers not familiar with the notion of a vector relative degree of a

nonlinear system are referred to [28, Chap. 5] and [40]. In essence, thelast assumption implies that (1) is locally static input-output feedbacklinearizable. A key ingredient in the further investigations will be thenotion of a transverse normal form of a path-following problem,which is in essence a nonlinear input-output normal form tailored topath-following problems, see also [5], [7].

Lemma 3 (Local existence of a transverse normal form):Consider system (1). Suppose that Assumption 6 holds andthat in the timing law (3) r from (13) is used. Then the followingstatements hold for all (x, z)T ∈ Nx ×Z with x ∈ int(X ):

i) The augmented system (4) has a well-defined vector relativedegree r = (r1, . . . , rny , r)

T .ii) There exists a local diffeomorphism Φ : Rnx × Rr → Rρ ×

Rnx+r−ρ, (x, z) 7→ (ξ, η) such that (4) is equivalent to atransverse normal form

ξi = Iri−1ξi +

(0ri−1,1

αi(ξ1, . . . , ξny , η, u, v)

), i ∈ {1, . . . , ny}

(14a)

η = β(ξ, η, u, v) (14b)

with

ξ =

(e1, e1, . . . , e

(r1−1)1︸ ︷︷ ︸, . . . , eny , . . . , e

(rny−1)ny︸ ︷︷ ︸ )T

,

ξ1 ξny

whereby ξ ∈ Rρ and ρ =∑nyi=1 ri, η ∈ Rnx+r−ρ.

Proof: The proof mainly exploits the fact that the dynamics ofx and z are only coupled via the output of (4). We show how theLie derivatives of the output of the augmented system (4) can beobtained, and thereby we proof part i) of the Lemma. Part ii) followsdirectly by results given in [28], [40].

Using the simple change of coordinates χ = (x, z)T , ν =(u, v)T , µ = (e, θ)T system (4) can be written as

χ = φ(χ) +

nu+1∑j=1

ωj(χ)νj (15a)

µ = ψ(χ). (15b)

The vector fields φ : Rnx+r → Rnx+r, ωj : Rnx+r → Rnx+r, ψ :Rnx+r → Rny+1 follow directly from (4).

Calculating the Lie derivatives of ψi(χ) = hi(x) − pi(z1), i ∈{1, . . . , ny} with respect to νj , j ∈ {1, . . . , ny + 1} yields

Lωj Lkφ ψi(χ) =

{Lgj Lkf hi(x) j ∈ {1, . . . , ny}LE Lk

Ipi(z1) j = ny + 1

. (16a)

Assumption 6 implies that Lgj Lkf hi(x) = 0 for k ∈ {1, . . . , ri −2}, i, j ∈ {1, . . . , ny}. From (4) it follows that LE Lk

Ipi(z1) = 0

for k ∈ {1, . . . , r − 2}, i ∈ {1, . . . , ny}.Due to Assumption 6 it is clear that for k = ri − 1 and at least

one j ∈ {1, . . . , ny}

Lωj Lkφ ψi(χ) = Lgj Lkf hi(x) 6= 0. (16b)

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Fig. 2: Geometric interpretation of the transverse normal form (14).

Now consider the case i = ny + 1, i.e., consider the outputµny+1 = θ = z1 of (4). Since this output is only influenced byνny+1 = v, it follows that

Lωj Lkφ ψny+1(χ) =

0, j ∈ {1, . . . , ny + 1}, k ∈ {1, . . . , r − 2}0, j ∈ {1, . . . , ny}, k = r − 1

1, j = ny + 1, k = r − 1.(16c)

The conditions (16a–c) imply that the decoupling matrix of (15) hasthe following structure

A(χ) =

Lg1 Lr1−1

f h1(x) . . . Lgny Lr1−1f h1(x) ∗

.... . .

......

Lg1 Lrny−1

f hny (x) . . . Lgny Lrny−1

f hny (x) ∗0 . . . 0 1

=

(A(x) ∗01,ny 1

).

Note that this matrix is an upper triangular block matrix, wherebythe decoupling matrix of the original system (1) appears as upper leftblock. The stars in the upper right block replace Lωj Lkφ ψi(χ) fori ∈ {1, . . . , ny}, j = ny + 1, k = ri − 1. These terms are either 0(for ri < r) or 6= 0 (for ri = r). However, they do not affect the rankof A(χ). Due to Assumption 6 we know that the upper left blockA(x) of this matrix has full rank in an open neighborhood Nx of x.Thus A(χ) has full rank on Nx × Z . From this and (16) it followsthat part i) of the lemma is verified, i.e., on Nx ×Z the augmentedsystem (4) has a vector relative degree of r = (r1, . . . , rny , r)

T .In order to obtain the diffeomorphism that maps the system to

a (local) transverse normal form one picks ξi = Lkφ ψi for k ∈{0, . . . , ri − 1}, i = {1, . . . , ny} as new coordinates. This specifiesρ coordinates ξ with ρ =

∑nyi=1 ri ≤ nx + r, which are transverse

to the path manifold IP characterized by ξ = 0, cf. Figure 2. Theexistence of additional nx + r − ρ independent coordinates followsdirectly from the fact that the augmented system has a well-definedvector relative degree, cf. [28, Prop. 5.1.2] or [40].5 This finishes the

5For instance, one can pick r coordinates by the identity η1 = z. This wayit only remains to pick nx − ρ coordinates, which are directly related to theinternal dynamics of (1a) with respect to the output (1b). This situation isalso illustrated in Figure 2. We refer to [36, Chap. 4] for an example showingthat also η1 6= z might be helpful in some cases.

proof.

Remark 5 (Transverse normal forms): Note that the directionsξ ∈ Rρ are composed of the path error e = h(x) − p(z1) and itsderivatives, i.e., these directions point away from the path manifold.A graphical interpretation of this situation is depicted in Figure 2.More precisely, these directions are transverse—i.e., orthogonal—tothe manifold of trajectories, which travel along the path IP . Thistransversality is the reason to denote (14) as a transverse normalform. It should be recognized that the directions η ∈ Rnx+r−ρ arenot specified. This implies that transverse normal form descriptionsare usually not unique. Additionally, it is worth to be mentionedthat Assumption 6 is only sufficient but not necessary for theexistence of a transverse normal form. If dimu > dim y, onecan use ideas from [28, Chap. 5] to derive the normal form. Ifdimu < dim y, the situation is more complicated. In the specialcase dim y = dimu + 1 one can attempt to use the virtual input vto achieve dim y = dim(u, v)T . An example of a transverse normalform for a system with dimu = 1 and dim y = 2 can be found in[36, Chap. 4.3].

Note that such descriptions of path-following problems wereinitially proposed in [5], [7]. Thus, results similar to Lemma 3 can,for instance, be found in [7]. However, our approach slightly differsfrom these results: We work with a known path parametrizationp : R → Rny from (2) and the corresponding augmented system(4), while the results in [7] consider implicitly defined paths whereno parametrization is known. The consideration of the path parameterstates z in the augmented system (4) allows the description of thereference motion along of the path.

Beyond these structural observations the lemma reveals that outputpath-following implies the stabilization of a specific manifold—theso-called path manifold, denoted as IP in Figure 2—in the statespace. This manifold is locally characterized by the condition ξ = 0.Hence, it is not surprising that the computation of terminal regionsand end penalties satisfying Theorem 1 is in general challenging.In essence, such a computation implies to solve at least locally amanifold stabilization problem in the presence of input and stateconstraints.6 One may wonder whether there is any hope to compute

6At this point it is fair to ask for sufficient or necessary path-followabilityconditions. In other words, one may ask for conditions ensuring that a systemcan be steered along a path exactly. This question is beyond the scope of thispaper. Results in this direction for unconstrained and constrained systems canbe found in [36], [41].

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terminal penalties for the MPFC schemes (6) along the lines of [33],[38], i.e., based on a linearization of the augmented dynamics (4)around a specific point. This is in general difficult for two reasons:First, the constraints on the path parameter states z ∈ Z (7) implythat the final path point θ1 is not contained in the interior of Z .Thus ellipsoidal terminal regions based on a linearization of (4) or(14) at a single point of the state space are not well suited, since(θ1, 0, . . . , 0)T ∈ ∂Z implies that any ellipsoidal terminal regionwould shrink to a single point in the directions associated with thepath parameter state z. Second, the structure of the internal dynamicsof the transverse normal form (14) has to be taken into account.Thus it is difficult to state a general procedure for the computationof suitable terminal regions.

Remark 6 (MPFC without terminal constraints): To reduce thecomputational burden one might also ask for conditions which ensurepath convergence without terminal constraints. For NMPC for set-point stabilization such conditions are discussed i.a. in [42], [43].For predictive path following two major issues arise if one attemptsto drop the terminal constraint:

i) The guarantees of recursive feasibility in the presence of stateconstraints are in general lost, respectively, rather difficult toenforce. As a remedy one could drop the state constraints of thereal system (1). The forward motion requirement ii) of Problem1, however, inevitably leads to constraints of the virtual states z.Thus one might need to drop the forward motion requirement aswell as and merely require convergence of the path parameter,i.e., lim

t→∞‖θ(t)− θ1‖ = 0.

ii) Note that the presence of the terminal constraints, which area compact set E ⊆ X , implies that all states remain boundedduring the application of the MPFC scheme. This is due to theassumed continuity of solutions (Assumption 2) and the factthat at the end of each prediction over a finite horizon theaugmented state (x, z)T has to be inside the compact terminalset. If neither (compact) state constraints x ∈ X nor a compactterminal constraint are considered, extra care has to be takenin order to ensure boundedness of the states. Taking Lemma 3into account, it is clear that the states contained in the zerodynamics of the augmented system (4) with respect to theoutputs e, θ might cause difficulties, for instance, due to non-minimum phase behavior. Preliminary results presented in [44]indicate that via structural assumptions on the system dynamicsthese issues might be avoided.

Remark 7 (Generalized cost functions for MPFC): In the view ofthe transverse normal forms of Lemma 3 one could as well penalizenot only the path-following error e but also its time derivatives inthe cost function F as for instance considered in [18], [19], [44].However, note that to enforce path convergence it suffices to rely onAssumption 4, i.e., lower boundedness of F by ψ(‖e, θ − θ1‖).

Finally, it should be mentioned that the rewriting the augmentedsystem (4) in a transverse normal form is not necessary to designan MPFC scheme. Indeed for many examples system descriptionsin transverse coordinates exist only locally. However, transversecoordinates are very helpful in the sense that they allow to gaininsight to the geometry of path-following problems and their use oftensimplifies the design of terminal regions. In the next section and inAppendix A we draw upon an example from robotics to demonstratethis.

V. EXAMPLE: FULLY ACTUATED ROBOT

To illustrate the proposed MPFC scheme we consider a fullyactuated planar robot with two degrees of freedom. Without friction

and external contact forces the dynamics of such a robot are givenby (

x1

x2

)=

(x2

B−1(x1) (u− C(x1, x2)x2 − g(x1))

)(17a)

y = x1 (17b)

yca = hca(x1) (17c)

Here x1 = (q1, q2) ∈ R2 is the vector of joint angles, x2 =(q1, q2) ∈ R2 is the vector of joint velocities. B : R2 → R2×2 andC : R4 → R2×2 describe the dependence of the inertia on the jointangles and the dependence of centrifugal and Coriolis forces on jointangles and velocities, respectively. The function g : R2 → R2 modelsthe effect of gravity. The model details are provided in Appendix A.

The output y = x1 denotes the space of joint angles, the outputyca = hca(x1) is the position of the robot tool in Cartesiancoordinates. The inputs u = (u1, u2)T are the torques applied toeach joint. We consider box constraints on states and inputs

U ={u ∈ R2 | ‖u‖∞ ≤ u

}(18a)

X ={x = (x1, x2) ∈ R4 | ‖x2‖∞ = ‖(q1, q2)‖∞ ≤ ¯q

}(18b)

whereby u = 4000 Nm and ¯q = 32π rad/s.

The considered path-following task is described in the joint space.The path is specified via the parametrization p : [θ0, θ1]→ Rny

p(θ) =(θ − π

3, ω1 sin(ω2(θ − π

3)))T (19)

where θ0 = −5.3, θ1 = 0, ω1 = 5, ω2 = 0.6.

A. Simulation Results

The cost function for the MPFC scheme is chosen according toRemark 7, i.e., we penalize the path error e and it stime derivativee.

F (e, e, θ, u, v) =∥∥∥(e, e, θ)T

∥∥∥2

Q+∥∥∥(u− u, v)T

∥∥∥2

R(20)

whereby Q = diag(105, 105, 10, 10, 5) and R =diag(10−3, 10−3, 10−4). This way we also satisfy Assumption4. The offset u = (263.0,−262.5)T = g(p(0)) corresponds to thetorque required to keep the robot at the final path point p(0). InAppendix A we show how to derive the following terminal regionfor the augmented system (23)

E ={

(x, z) ∈ R6 | (ξ, η) = Φ(x, z), ξTPξξ ≤ 3.13, η ∈ Eη}

(21a)

Eη ={z ∈ R2 | z1 ∈ [−5.3, 0], z2 ∈ [0, 0.4], nT z ≤ 0

}(21b)

with n = (0.78, 0.63)T . The virtual states z are restricted to apolyhedral terminal region Eη which is sketched in Figure 4 inAppendix A. Additionally, in (21a) the directions of the augmentedstate (x, z) that are transverse to the path manifold are restricted toan ellipsoidal terminal region, whereby Pξ is from (34). Furthermore,we show in Appendix A that E (21) satisfies the conditions of Lemma1. Thus, according to Proposition 1 we consider the trivial terminalpenalty E(t, x, z) = 0.

The simulations are performed with the following parameters: Thevirtual input v is restricted to V = [−50, 50]. The prediction horizonis set to T = 0.75s, the sampling time is δ = 0.005s and OCP (6)is solved repeatedly with a direct multiple shooting implementationusing 20 shooting intervals [45].

Figure 3a presents simulations results for the initial conditionx(0) = (−5.86, 2.43, 0, 0)T , z(0) = (−5.3, 0)T . The upper left sideshows the time evolution of the joints x1(t) = (q1(t), q2(t)) in black

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1 2 3 4

−6

−3

0

3

6

t [s]

x1,2

[rad]

Joint angles

p1(z1)x1

p2(z1)x2

1 2 3 4

−6

−3

0

3

6

t [s]

x3,4

[rad/s]

Joint velocities

x3,4 minx3,4 maxx3x4

1 2 3 4

−4000

−2000

0

2000

4000

t [s]

u1,2

[Nm]

Torques

u1,2 minu1,2 maxu1u2

1 2 3 4

−10

−5

0

5

10

t [s]

z 1,2,v 5[–]

Virtual states

vmin/5

vmax/5

v/5z1z2

(a) Simulated closed-loop trajectories for 2-DoF robot.

−7 −5.5 −4 −2.5 −1

−5

−3

−1

1

3

x1 [rad]

x2[rad]

x1 − x2 plane

−0.6 −0.2 0.2 0.6

−0.8

−0.4

0

0.4

0.8

yca,1 [m]

yca,2[m

]

Cartesian position

(p1(θ), p2(θ))

(x1, x2)

hca (p1(θ), p2(θ))

hca (x1, x2)

(b) Path convergence in joint space (left) and Cartesian output space (right).

Fig. 3: Simulation results for 2-DoF robot.

color and the reference p(z1(t)) in gray color. The joint positionsconverge rapidly to the reference. The upper right side depicts thecorresponding joint velocities and their constraints. In the lower rightside the virtual states z1 = η1, z2 = η2 and the virtual input v areplotted. One can observe that the path parameter moves forward tothe end of the path at θ = z1 = 0. Also note that the MPFC schemeuses the virtual input v to adjust the speed along the reference. Theinput torques are shown in the lower left side of Figure 3a. Bothinputs satisfy the constraints.

In Figure 3b the path convergence for different initial conditionsis depicted. On the left side the plane of joint angles x1 = (q1, q2) isplotted. The black arrows indicate the direction of movement of therobot. On the right side it is shown how the solutions for differentinitial conditions converge to the image of the path in the Cartesianoutput space defined via (22d). One can see that the proposed MPFCscheme ensures path convergence for a range of initial conditions.Finally, we conclude that the conditions of Theorem 1 can beused to design predictive path-following controllers. In presence ofconstraints on states and inputs the MPFC scheme enforces pathconvergence and convergence on the path.

VI. CONCLUSIONS

This paper has presented a predictive control scheme for con-strained path-following problems with and without velocity assign-ment that guarantees convergence subject to sufficient convergence

conditions based on terminal regions and end penalties. In contrast togeometric or backstepping approaches to path following the proposedmodel predictive path-following control scheme allows to handleconstraints on states and inputs as well as nonlinear dynamics andreference paths. Furthermore, we have established structural insightsinto path-following problems via transverse normal forms, whichallow simplified computation of terminal regions and end penalties.

APPENDIX ACOMPUTATION OF A TERMINAL REGION FOR THE EXAMPLE

A. Model Details for the Robot Example

The terms B,C, g, hca of (17) are as follows

B(q) =

(b1 + b2 cos(q2) b3 + b4cos(q2)

b3 + b4cos(q2) b5

)(22a)

C(q, q) = −c1 sin(q2)

(q1 q1 + q2

−q1 0

)(22b)

g(q) =(g1 cos(q1) + g2 cos(q1 + q2), g2 cos(q1 + q2)

)T(22c)

hca(q) =

(l1 cos(q1) + l2 cos(q1 + q2)

l1 sin(q1) + l2 sin(q1 + q2)

). (22d)

The system parameters are listed in Table I, cf. [46].

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B. Problem Description in Transverse Normal Form

It is easy to see that (17) has a global vector relative degree r =(2, 2)T with respect to the output y = x1. Thus, we use as pathparameter dynamics an integrator chain of length two and obtain theaugmented system descriptionx1

x2

z

=

x2

B−1(x1) (u− C(x1, x2)x2 − g(x1))

I2z + E2v

(23a)

e = x1 − p(z1) (23b)

θ = z1. (23c)

We want to map these augmented dynamics into a transverse normalform. Following along the lines of the proof of Lemma 3 we obtainthe coordinate transformation Φ : R4 × R2 → R4 × R2 and itsinverse Φ−1 : R4 × R2 → R4 × R2

Φ : ξ1 = x1 − p(z1), ξ2 = x2 −∂p

∂z1z2, η = z (24a)

Φ−1 : x1 = ξ1 + p(η1), x2 = ξ2 +∂p

∂η1η2, z = η. (24b)

Observe that to simplify the later derivations, we have chosen adifferent ordering of the ξ-variables compared to (14). Furthermore,note that only the virtual states z appear in η. This is due to the factthat (17a) has a state dimension of nx = 4 and the vector relativedegree (17a) r = (2, 2)T with respect to y = x1. Thus the robotdynamics (17a) do not have internal dynamics with respect to (17b).

Due to its simple structure and the assumptions on the pathparametrization p(θ) it is easy to see that Φ : R4 ×R2 → R4 ×R2

is a global diffeomorphism. Using Φ it is straightforward to rewritethe augmented robot dynamics (23) into the transverse normal form.We obtain ξ1ξ2

η

=

ξ2

α(ξ, η, u, v)

I2η + E2v

(25a)

e = ξ1 − p(η1) (25b)

θ = η1, (25c)

whereby the vector field α : R4 × R2 × R2 × R→ R2 is

α(ξ, η, u, v) = B−1(ξ1, η1)

(u− C(ξ, η)

(ξ2 −

∂p

∂η1η2

)

− g(ξ1, η1)

)− ∂2p

∂η21

(η2)2 − ∂p

∂η1v. (26)

C. Design of a Terminal Region

We design a terminal control laws for the augmented dynamicsin transverse normal form (25). Note that in (23) as well as in (25)the dynamics of z = η are not influenced by the other states. Thuswe first design a terminal control and terminal region for the η-dynamics and subsequently consider the transverse dynamics. Recallthat the path parameter dynamics η = I2η+E2v are simply a doubleintegrator and the state constraint Z (7) is a polytope Z = {η ∈

TABLE I: Robot parameters [46].

b1 200.0 [kg m2/rad] b2 50.0 [kg m2/rad]b3 23.5 [kg m2/rad] b4 25.0 [kg m2/rad]b5 122.5 [kg m2/rad] c1 −25.0 [Nms−2]g1 784.8 [Nm] g2 245.3 [Nm]l1 0.5 [m] l2 0.5 [m]

Fig. 4: Terminal region for the path parameter dynamics.

R2 | η1 ∈ [θ0, 0], η2 ≥ 0} with η = z. A sketch of the stateconstraint Z is shown in Figure 4. Part ii) of Problem 1 requires that aterminal control law achieves limt→∞ η(t) = limt→∞ z(t) = (0, 0).In other words, the path parameter state η should converge to theorigin and the constraint η(t) ∈ Z implies that θ = η1 ∈ [θ0, 0] andθ = η2 ≥ 0. Since the origin is contained in the boundary of theZ , ellipsoidal terminal regions for the η part of (25) would shrinkto a point. Thus we aim on constructing a polytopic terminal region,which is rendered positively invariant by a linear feedback vE = Kη.

The coefficients of Kη should satisfy k1, k2 < 0 in order enforceasymptotic convergence to the origin and k2

2 > −4k1 to avoidoscillations. Furthermore, it is easy to verify that the eigenspacesof I2 +E2Kη corresponding to such a choice for Kη lie in the 2nd

and 4th quadrant of the η1 − η2 phase plane. Exemplarily this isdepicted by the blue lines in Figure 4. We use

vE = Kηη, Kη = (k1, k2),

k1, k2 < 0, k22 > −4k1, k2 ≤ −k1θ0

(¯θ)−1

< 0 (27)

as a terminal feedback for η. The additional condition k2 ≤ −k1θ0/¯θ

ensures that the initial velocity vector of any closed-loop solutionstarting on the line η =

(α,

¯θ), α ∈ [θ0, 0] points towards the η1-

axis. Furthermore, it is easy to verify that for such a choice of Kη

all solutions starting somewhere in the interval [θ0, 0] on the negativeη1 axis converge to the origin with θ = η2 > 0. Additionally, wehave for any choice of Kη that the solutions starting on the positiveθ = η2 axis will leave the state constraint set Z . Based on theseconsiderations we choose the terminal region for the η-dynamics as

Eη :={η ∈ R2 | η1 ∈ [θ0, 0], η2 ∈

[0,

¯θ], nT1 η ≤ 0

}. (28)

This terminal region is sketched in green color in Figure 4. Here,n1 is the normal vector corresponding to the upper eigenspace ofI2 +E2Kη . The verification of positive invariance of Eη with respectto η = (I2+E2Kη)η follows directly from the considerations above.The boundary 0 ≤ η2 ≤ ¯

θ is introduced to Eη in order to simplifythe design of a terminal region for the ξ-dynamics.

We proceed with the design of a suitable feedback for the ξ-partof the transverse dynamics (25). As a terminal feedback we use

uE(ξ, η) = C(ξ, η)

(ξ2 −

∂p

∂η1η2

)+ g(ξ1, η1)

+B(ξ1, η1) (Kξξ + p(η1(t))) . (29)

It is easy see to that this feedback achieves global exact staticfeedback linearization of (25). More precisely it achieves globaltransverse feedback linearization, cf. [5], [7]. The term Kξξ willbe used to stabilize the path manifold. The p(η1(t) part can beunderstood as a feedforward control. Using this feedback the ξ-partof (25) is governed by ξ = (Aξ +BξKξ)ξ. W.l.o.g. we assume that

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we have designed a stabilizing gain matrix Kξ and that

V (ξ) = ξTPξξ, Pξ > 0 (30)

is a corresponding Lyapunov function.Now, we are ready to derive a terminal region Eξ ⊂ R4 for the

transverse part of (25). The main idea is to bound the norm of thefeedback (29) from above and to obtain the terminal region Eξ ⊂ R4

as a level set of V (ξ). Due their structure the terms B : R2 →R2×2, C : R2 × R2 → R2×2 and g : R2 → R2 from (22a-f) can bebounded from above by constants

∀x ∈ X : ‖B(x1)‖ ≤ B, ‖C(x1, x2)‖ ≤ C, ‖g(x1)‖ ≤ g.

These bounds also hold in in (ξ, η) coordinates. To bound ‖p(z1(t))‖from above we restrict ourselves to the set Eη from (28). Since z = η,p(θ) ∈ C2 and Eη is compact, we obtain

∀η ∈ Eη : ‖p(η1(t))‖ ≤∥∥∥∥∂2p

∂η21

η22 +

∂p

∂η1v

∥∥∥∥≤∥∥∥∥∂2p

∂η21

∥∥∥∥( ¯θ)2

+

∥∥∥∥ ∂p∂η1

∥∥∥∥ ∥∥∥k1θ0 + k2¯θ∥∥∥ =: ¯p.

Here, we have used that in the set Eη the η-dynamics are controlledvia v = Kηη. To simplify the further considerations we work withtightened constraints U ⊂ U , X ⊂ X

U ={u ∈ R2 | ‖u‖ ≤ u

}(31a)

X ={x = (x1, x2) ∈ R4 | ‖x2‖ = ‖(q1, q2)‖ ≤ ¯q

}(31b)

where in comparison to (18) the 2-norm is used instead of ‖ · ‖∞.Next, we apply the bounds derived before to the feedback uE from

(29). This yields

∀(ξ, η)T ∈ Φ(X × Eη

): ‖uE(ξ, η)‖ ≤ C ¯q+g+B

(¯p+ ‖Kξξ‖

).

We enforce that inside the terminal region to be determined, Eξ×Eη ,the tightened input constraint uE(ξ, η) ∈ U is satisfied. This is thecase if

∀(ξ, η)T ∈ Eξ × Eη : C ¯q + g + B(¯p+ ‖Kξξ‖

)≤ u.

Solving the last equation for ‖ξ‖ yields for all (ξ, η)T ∈Φ(X × Eη

):

‖ξ‖ ≤ u− C ¯q − g − B ¯p

B‖Kξ‖⇒ uE(ξ, η) ∈ U ⊂ U . (32)

Subsequently, we derive Eξ as a suitable level set of the Lyapunovfunction V (ξ) from (30). In general, the level set is

Eξ :={ξ ∈ R4 | ξTPξξ ≤ γ2

}.

The constant γ can be computed as follows

maximizeγ>0

γ (33a)

subject to

∀ξ ∈ Eξ : ‖ξ‖ ≤ u− C ¯q − g − B ¯p

B‖Kξ‖(33b)

∀ξ ∈ Eξ : ‖ξ2‖ ≤ ¯q − ¯p. (33c)

Here, ¯p is a bound on p(η1(t)) that can be obtained for η ∈ Eη ina similar fashion as ¯p. Given Kξ and Pξ this is a simplified versionof the (convex) problem to compute a maximum volume ellipsoidcontained in a convex set, cf. [47]. If ¯q − ¯p and the constant on theright side of (33b) are positive, problem (33) has a solution γ? > 0.This is the case if the input bound u and the bound ¯q are sufficientlylarge.

We use the model data from Table I and the path (19) to computenumerically the sets Eη and Eξ. The bound on η2 is set to ¯

θ = 0.4,and the feedback matrix for the η-dynamics is Kη = (−0.1,−1.33).This leads to the terminal constraint for η

Eη ={η ∈ R2 | η1 ∈ [−5.3, 0], η2 ∈ [0, 0.4], (0.78, 0.63)η ≤ 0

}.

The Lyapunov function (30) and the feedback matrix Kξ are com-puted via an LQR controller with Qξ = I4, Rξ = I2. This leadsto

Pξ =

(P1 P2

P2 P1

), Kξ = (P1, P2) (34)

with P1 = diag(1.73, 1.73) and P2 = I2. Solving (33) with thesevalues yields γ = 1.77. Thus the ellipsoidal part of the terminalregion is

Eξ ={ξ ∈ R4 | ξTPξξ ≤ 3.13

}.

Rewriting the terminal constraints in (x, z) coordinates yields

E ={

(x, z) ∈ R6 | (ξ, η) = Φ(x, z), ξTPξξ ≤ 3.13), η ∈ Eη}.

(35)

D. Derivation of a Terminal Penalty

It remains to derive an end penalty such that the conditions ofTheorem 1 or Proposition 1 are satisfied.

For the MPFC controller we use the quadratic cost function F from(20), which can be written in ξ, η coordinates as F (ξ, η1, u, v) =‖(ξ, η1)T ‖2Q + ‖(u − u, v)T ‖2R. It is straightforward to see that forall (ξ0, η0)T ∈ E , ∀t ≥ t0 :

‖ξ(t, t0, ξ0|uE(·))‖ ≤ cξ(ξ0, η0)e−αξ(t−t0)

‖η(t, t0, η0|vE(·))‖ ≤ cη(ξ0, η0)e−αη(t−t0),

whereby cξ(ξ0, η0) and cη(ξ0, η0) are bounded from above by finitenumbers. In other words, inside the terminal region (35) the applica-tion of the terminal control law (29) leads to exponential convergenceof the transverse directions ξ ∈ R4 and the path parameter stateη ∈ R2. Furthermore, it is clear that vE(t) = Kηη(t) is alsoconverging exponentially to zero. Using these bounds on ξ(t), η(t)and vE(t) we obtain that the solutions driven by the terminal feedback(29) satisfy

‖uE(ξ(t), η(t))− u‖ ≤ ‖C(x1(t), x2(t))x2(t)‖︸ ︷︷ ︸≤ Ccx2e

−αx2 (t−t0)

+ ‖B(x1(t)) (Kxξ(t) + p(η1(t))) ‖︸ ︷︷ ︸≤ B

(‖Kη‖cηe−αη(t−t0)+cpe

−αp(t−t0))

+ ‖g(x1(t))− g(p(0))‖︸ ︷︷ ︸≤ cge

−αg(t−t0)

The bound on the first term follows from x2 = ξ2 − p and‖p(t)‖ ≤ ‖ ∂p

∂θ‖cθe−αθ(t−t0). The bound on the second term follows

in a similar fashion. The estimate from above on ‖g(x1)− g(p(0))‖is more complicated: Note that x1 = ξ+p(η1). For p : [θ0, 0]→ R2

from (19) one can show that exponential convergence of η to 0implies exponential convergence of p(η1) to p(0). Using this wesee that for t→∞ also the state x1 converges exponentially to p(0)since x1 = ξ + p(η1). Finally, we use that in g : R2 → R2 from(22c) only cos-functions appear, and conclude that g(x1)− g(p(0))converges exponentially to 0. Since all the arguments of F convergeexponentially to 0 and F is quadratic, we see that the terminal region(35) and the terminal controls (27, 29) satisfy the assumptions ofLemma 1. In other words, E from (35) and the trivial terminal penaltyE(t, x(t), z(t)) = 0 allow applying Proposition 1.

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ACKNOWLEDGMENT

The authors would like to thank Friedrich von Haeseler from theOtto-von-Guericke University Magdeburg for valuable feedback anddiscussions on transverse normal forms.

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